3 THE MODULUS FUNCTION ON A GROUP 265
contradiction, by first showing that the sequence (Ap) may be assumed to have a limit
A' <= BO , and then using (c) above to prove that A' contains the frontier S0 of B0 ;
finally remark that £S0 + JS0 — B0 .)
(e) If A is any compact subset of R", show that
An(A)<2-»Vn(S(A)y
where the diameter 8(A) is relative to the Euclidean distance, and Vrt is the Lebesgue
measure of the ball Bn (*' Bieberbach's inequality")- (Use (d) and the inequality
10. With the notation of Problem 7, for any compact subset A of Rn the upper (resp.
lower) Minkowski area of A is defined to be the number
a+(A) = lim sup (An(Vr(A)) - An(A))/r
r-*0
(resp.
or(A) = lim inf (An(Vr(A)) - An(A))/r)
r-*0
where r tends to 0 through positive values.
(a) Give an example of a compact set A such that a+(A) = 1 and a "(A) = 0. (Take
n — 2, and consider a union of finite sets of the form ck -f Afc , where the sequence
(cfc) tends to 0, and for each k the set A* is a product Ifc x Jfc, where I*, J* are finite
sequences of the form (pak)Q^p^rk, (gbj)o^q^sk. Choose the sequences (cfc), (ak), (bk),
(rk), (sk) appropriately.)
(b) Show that
or(A) ^/
(the isoperimetric inequality). (Use the Brunn-Minkowski inequality.)
(c) Let A be a compact convex set (Section 8.5, Problem 8). Show that, for each
compact convex set B, the function of £
is convex for 0 <£ £ < + co, by using the Brunn-Minkowski inequality and the relations
(a + ]8)A = aA H- 0A, (a 4- ]8)B = aB 4- /?B for all real numbers a, ]8. Deduce that
a+(A) = a "(A) (use Problem 8 of Section 8.5). The number a(A) = a+(A) = a~(A)
is called the Minkoswki area of A.
(d) If the compact convex set A is contained in R"~ *, then a(A) = An_i(A).
(e) If 0 is an interior point of a compact convex set A <= R", then
wrf1An(A) ^ a(A) ^ nrg1Xn(A)9
where rt is the maximum of the radii of closed balls with center 0 and contained in A,
and re is the minimum of the radii of closed balls with 0 which contain A. (Use the
inequalities
11. Let G be a compact group, h a left invariant distance on G which defines the topology
of G (12.9.1) and JK, a Haar measure on G. Show that
is a bi-invariant distance on G, equivalent to h.hat 8(A + B) ^ 8(A) + 8(B).